As usual, let \(K_n\) , \(P_n\)  and \(K_{n_1,n_2,\cdots ,n_k}\) , where \(\sum \limits _{i=1}^kn_i=n\) , denote the complete graph, the path and the complete \(k\) -partite graph with order \(n\) , respectively. The girth of \(\Sigma =(G,\sigma )\) , short for \(g\) , is the girth of its underlying graph \(G\) , namely, the length of a shortest cycle in \(G\) . Let \(D(G)=(d_{uv})_{n\times n}\)  be the distance matrix of a connected graph \(G\) . Yu [1]} investigated the least distance eigenvalue of a graph, and determined all connected graphs
with \(\lambda _n(D(G))\in [-2.383,0]\)  among graphs with \(n\)  vertices. Lin, Hong, Wang and Shu [2]} extended the above result separately. The authors showed that \(\lambda _n(D(G))=-1\)  if and only if \(G\cong K_n\)  and \(\lambda _n(D(G))=-2\)  if and only if
\(G\cong K_{n_1,\cdots ,n_k}\) . And there does not exist a graph \(G\)  of order \(n\)  with \(-2<\lambda _n(D(G))<-1\) . Motivated by the papers, we pose the following problem:
